Mass-effect – Playing Mass Effect on multiple consoles

According to Wikipedia, you will need an Origin account for both the single-player and the multiplayer campaigns in ME3.

Customers purchasing PC editions of the game (retail or digital) will be required to install Electronic Arts' content delivery and digital rights management system, Origin. The Origin client (and an Origin account) is required to install, activate and run Mass Effect 3 on a PC for single and multi-player portions of the game.

This Bioware forum post by Chris Priestly gives a lot more detail around Origin and Mass Effect. Specifically, it speaks to your question:

Is constant Origin connection required or is it a single one off authentication when the game is first installed. Is there also a limit to the number of installations available? Mass Effect 3 will require a one time, single authorization for the single player game. There is no limit to the number of installs. Playing Mass Effect 3 Multiplayer will require a constant connection.

In addition to the iOS apps, which I noted in this answer:

Will I miss out on anything by not playing Mass Effect 3's multiplayer?

You also get a 2-day Xbox Live Gold card in with the game. If you use this time strategically (ie, as you're nearing the end of the game) you might be able to spend a weekend playing the multiplayer mode to boost your readiness score. (Remember that it decays over time so you can't just use it whenever and expect to keep the bonuses.)

If you're anything like me, you've probably also got a few more of these codes kicking around. I'm not sure what the limit is on their use, but you might be able to buy yourself just enough time to get by.

Also note that if you haven't already taken advantage of it, each console has a limited number of "free month" Xbox Live Gold offers. If you create a new profile, and sign up for Silver, there's a chance you'll be offered a month of Gold for free.

Any of these methods would likely get you enough time to make an impact on your overall score, but remember that you can still get the best ending by just playing single player.